Estimating covariances between financial assets plays an important role in risk management and portfolio allocation. Here, we show that from a Bayesian perspective market factor models, such as the famous CAPM, can be understood as linear latent space embeddings. Based on this insight, we consider extensions allowing for non-linear embeddings via Gaussian processes and discuss some applications.

In general, all these models are unidentified as the choice of coordinate frame for the latent space is arbitrary. To remove this symmetry we reparameterize the factor loadings in terms of an orthogonal frame and its singular values and provide an efficient implementation based on Householder transformations. Finally, relying on results from random matrix theory we derive the parameter distribution corresponding to a Gaussian prior on the factor loadings.